The Symmetric Sugeno Integral

TL;DR

This paper introduces a symmetric extension of the Sugeno integral for negative numbers within an ordinal framework, incorporating a new algebraic structure and M"obius transform, similar to the symmetric Choquet integral.

Contribution

It presents the first symmetric Sugeno integral for negative numbers, extending the ordinal-based Sugeno integral with a novel algebraic structure and M"obius transform.

Findings

Defined negative numbers on a linearly ordered set.

Developed a new algebra close to real numbers.

Introduced the symmetric Sugeno integral with properties similar to the symmetric Choquet integral.

Abstract

We propose an extension of the Sugeno integral for negative numbers, in the spirit of the symmetric extension of Choquet integral, also called \Sipos\ integral. Our framework is purely ordinal, since the Sugeno integral has its interest when the underlying structure is ordinal. We begin by defining negative numbers on a linearly ordered set, and we endow this new structure with a suitable algebra, very close to the ring of real numbers. In a second step, we introduce the M\"obius transform on this new structure. Lastly, we define the symmetric Sugeno integral, and show its similarity with the symmetric Choquet integral.